As the Olympic park waits for padlocks, where is the legacy? Politicians have been bickering because they don't know what to do. Boris Johnson wants two hours of sports in schools daily, the better to produce "semi-naked women … glistening like wet otters". Jeremy Hunt wants more competitive sport in schools, which shows how little he knows about pleasing children. David Cameron, I suspect, merely wants to halt allegations of him "hexing" the team with his actual physical presence.

He is now sitting in his Team GB polo shirt in No 10, insisting he junked Labour's compulsory two hours of sport a week because schools were using it to push "Indian dance" and other things he doesn't understand. The political class didn't see the medals coming, possibly because, as one campaigner told me, politicians hate sport as they aren't very good at it. Jocks frighten them.

How about equality? If the Olympics feel like a dreamscape, it is because they are partially class- and gender-free, and so alien to our ordinary hierarchies that we barely recognise them. Even so, they have exposed sexism in sport because the female athletes have identified it. Cyclist Lizzie Armitstead, who won Britain's first medal, said: "It's overwhelming and frustrating, the sexism I've experienced in my career. It's the obvious things – the salary, media coverage…" (She didn't mention what it is like to be a glistening otter wriggling in slow-mo.)

Both sponsorship and TV coverage are agonisingly low for sportswomen. According to the Women's Sport and Fitness Foundation, women got only 0.5% of sponsorship and 5% of TV coverage between January 2010 and August 2011, even though the women's football World Cup final was the most tweeted about event on the planet in 2011. The same year the BBC forgot to shortlist any women for their sports personality prize. The situation is so wretched that UK Sport, the body that distributes funds to athletes, has asked for at least one in four seats on governing bodies to be held by women, and is considering financial penalties to ensure it. Female politicians and businesswomen will identify; to be a female Olympian is harder.

The sexism is everywhere. Frankie Boyle, who can always be relied upon to project his self-hatred on to passing women, said: "I worry Rebecca Adlington will have an unfair advantage in the swimming by possessing a dolphin's face."Nicola Adams's victory exposed a backstory in which she could not box competitively in childhood, due to the impenetrable mysteries of the menstrual cycle. This is a sport where women were told to wear skirts in the ring as recently as March (to differentiate them from men, you see) and where former world champion Amir Khan warned female boxers: "When you get hit it can be very painful." Really? Who knew?

The female Japanese footballers flew to London in economy, while the men went business; the fact the women are ranked third in the world, and the men 20th, was irrelevant; it was, literally, downgrade. NBC, the US network, aired a slow motion montage of female athletes jumping, sprinting and bouncing to the kind of music you hear in 1970s porn films, with close-ups of their pants. In response, the Sun decided it was annoyed by this casual objectification of women and called it "porny". It even quoted the feminist website Jezebel.

This is a testament to the wild power of nationalism; if the Sun is minding sexism, we are at a tipping point. This is the first Olympics where every competing nation has sent women and the first where women have competed in every sport, even if the Saudi Arabian runner Sarah Attar had to run 800m in a hijab.

Sponsors and broadcasting executives, enslaved by the idea that viewers don't want to watch women play sport, are thwarting progress. They should read the Women's Sport and Fitness Foundation's excellent report, Big Deal, which junks this fantasy – in 2010 61% of sports fans said they wanted more women on the screen. But the prejudice trickles down to schoolgirls who, lacking female athletes as role models and considering sport "unfeminine" because it does not conform to the necrophilic tendencies of fashion advertising – which only wants women to stop eating – lose interest; only 12% of girls exercise enough, compared to 24% of boys.

It is gratifying to watch women grow strong but I mistrust any activities that bring joy through osmosis. Surely that is the opposite of athleticism? More important is the possibility that the Olympics might encourage girls not to fall into obesity and hate the mirror. Competitive sport in schools, which the government thinks is important because politicians love to win, is unhelpful by itself; it rockets girls into the safer havens of bewilderment and jelly.

Politicians are now open to all sorts of possibilities. They should consider a ban on sugar advertising and a reinstatement of the School Sports Partnership, with emphasis on the needs of the more fearful children, such as future politicians. They could go wild with inspirations and outcomes. By autumn the Olympic park will be empty. Many more young, strong women. Now that is a legacy.